Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $q = \dfrac{7k + 7}{9k - 54} \times \dfrac{k^2 - 12k + 36}{k - 6} $
Answer: First factor the quadratic. $q = \dfrac{7k + 7}{9k - 54} \times \dfrac{(k - 6)(k - 6)}{k - 6} $ Then factor out any other terms. $q = \dfrac{7(k + 1)}{9(k - 6)} \times \dfrac{(k - 6)(k - 6)}{k - 6} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac{ 7(k + 1) \times (k - 6)(k - 6) } { 9(k - 6) \times (k - 6) } $ $q = \dfrac{ 7(k + 1)(k - 6)(k - 6)}{ 9(k - 6)(k - 6)} $ Notice that $(k - 6)$ appears twice in both the numerator and denominator so we can cancel them. $q = \dfrac{ 7(k + 1)\cancel{(k - 6)}(k - 6)}{ 9\cancel{(k - 6)}(k - 6)} $ We are dividing by $k - 6$ , so $k - 6 \neq 0$ Therefore, $k \neq 6$ $q = \dfrac{ 7(k + 1)\cancel{(k - 6)}\cancel{(k - 6)}}{ 9\cancel{(k - 6)}\cancel{(k - 6)}} $ We are dividing by $k - 6$ , so $k - 6 \neq 0$ Therefore, $k \neq 6$ $q = \dfrac{7(k + 1)}{9} ; \space k \neq 6 $